Problem about the definition of Euclidean domain

From my sci.math post on 2009/7/2: The property $\rm V(a) \le V(ab)$ needn't be assumed in order to deduce all of the basic properties of Euclidean domains. It is true that any Euclidean function can be normalized to satisfy said property by defining $\rm\:v(a) = min\: V(aD^*),\ D* = D\backslash0.\:$ This is so well-known it is even in the Wikipedia http://en.wikipedia.org/wiki/Euclidean_domain Compare also the analogous Dedekind-Hasse criterion for a PID. And be sure to see this paper[1]. It gives an in-depth study and comparison of a dozen different definitions/axioms for Euclidean rings.

[1] Euclidean Rings. A. G. Agargun, C. R. Fletcher
Tr. J. of Mathematics, 19, 1995, 291 - 299.